1. 2.1 Graphs of Sine and Cosine Functions

2. Learning Objectives Understand the graph of y = sin x.
Graph variations of y = sin x.
Understand the graph of y = cos x.
Graph variations of y = cos x.
Use vertical shifts of sine and cosine curves.
Model periodic behavior.

3. The Graph of y = sinx

4. Some Properties of the Graph of y = sinx
The domain is
The range is [−1, 1].
The period is
The function is an odd function:

5. Graphing Variations of y = sinx
Identify the amplitude and the period.
Find the values of x for the five key points—the three x-intercepts, the maximum point, and the minimum point. Start with the value of x where the cycle begins and add quarter-periods—that is,
—to find successive values of x.
Find the values of y for the five key points by evaluating the function at each value of x from step 2.
Connect the five key points with a smooth curve and graph one complete cycle of the given function.
Extend the graph in step 4 to the left or right as desired.

6. Example 1: Graphing a Variation of y = sinx (1 of 4)
Determine the amplitude of
Then graph
Step 1: Identify the amplitude and the period.
The equation
is of the form
Thus, the amplitude is
This means that the
Maximum value of y is 3 and the minimum value of y is −3.
The period is

7. Example 1: Graphing a Variation of y = sinx (2 of 4)
Determine the amplitude of
Then graph
Step 2: Find the values of x for the five key points. To generate x-values for each of the five key points, we begin by dividing the period,
by 4. The cycle begins
at x1 = 0. We add quarter periods to generate x-values for each of the key points.

8. Example 1: Graphing a Variation of y = sinx (3 of 4)
Step 3: Find the values of y for the five key points.

9. Example 1: Graphing a Variation of y = sinx (4 of 4)
Determine the amplitude of
Then graph
Step 4: Connect the five key points with a smooth curve and graph one complete cycle of the given function.

10. Amplitudes and Periods
Amplitudes and periods The graph of y = A sin Bx, B > 0, has

11. Example 2: Graphing a Function of the Form y = AsinBx (1 of 5)
Determine the amplitude and period of
Then graph the function for
Step 1: Identify the amplitude and the period. The equation is of the form y = A sinBx with A = 2 and
The maximum value of y is 2, the minimum value of y is −2. the period of
tells us that the graph completes one
cycle from 0 to

12. Example 2: Graphing a Function of the Form y = AsinBx (2 of 5)
Step 2: Find the values of x for the five key points.
To generate x-values for each of the five key points, we begin by dividing the period,
by 4.The cycle begins
at x1 = 0. We add quarter periods to generate x-values for each of the key points.

13. Example 2: Graphing a Function of the Form y = AsinBx (3 of 5)
Step 3: Find the values of y for the five key points.

14. Example 2: Graphing a Function of the Form y = AsinBx (4 of 5)
Determine the amplitude and period of
Then graph the function for
Step 4: Connect the five key points with a smooth curve and graph one complete cycle of the given function.

15. Example 2: Graphing a Function of the Form y = AsinBx (5 of 5)
Determine the amplitude and period of
Then graph the function for
Step 5: Extend the graph in step 4 to the left or right as desired. We will extend the graph to include the interval

16. The Graph of y = Asin(Bx − C)
is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the circle is shifted from x = 0 to If
the shift is to the right. If
the shift is to the left. The number
is called the phase shift.

17. Example 3: Graphing a Function of the Form y = Asin(Bx − C) (1 of 5)
Determine the amplitude, period, and phase shift of
Then graph one period of the function.
Step 1: Identify the amplitude, the period, and the phase shift.

18. Example 3: Graphing a Function of the Form y = Asin(Bx − C) (2 of 5)
Step 2: Find the values of x for the five key points.

19. Example 3: Graphing a Function of the Form y = Asin(Bx − C) (3 of 5)
Step 3: Find the values of y for the five key points.

20. Example 3: Graphing a Function of the Form y = Asin(Bx − C) (4 of 5)
Step 3: Find the values of y for the five key points. [continued]

21. Example 3: Graphing a Function of the Form y = Asin(Bx − C) (5 of 5)
Step 4: Connect the five key points with a smooth curve and graph one complete cycle of the function

22. The Graph of y = cosx

23. Some Properties of the Graph of y = cosx
The domain is
The range is [−1, 1].
The period is
The function is an even function:

24. Sinusoidal Graphs
The graphs of sine functions and cosine functions are called sinusoidal graphs.
The graph of
is the graph of
with a phase shift of

25. Graphing Variations of y = cosx
The Graph of y = A cos Bx
The graph of y = A cos Bx, B > 0, has

26. Example 4: Graphing a Function of the Form y = AcosBx (1 of 6)
Determine the amplitude and period of
Then graph the function for
Step 1: Identify the amplitude and the period.

27. Example 4: Graphing a Function of the Form y = AcosBx (2 of 6)
Determine the amplitude and period of
Then graph the function for
Step 2: Find the values of x for the five key points.

28. Example 4: Graphing a Function of the Form y = AcosBx (3 of 6)
Step 3: Find the values of y for the five key points.

29. Example 4: Graphing a Function of the Form y = AcosBx (4 of 6)
Step 3: Find the values of y for the five key points. [continued]

30. Example 4: Graphing a Function of the Form y = AcosBx (5 of 6)
Determine the amplitude and period of
Then graph the function for
Step 4: Connect the five key points with a smooth curve and graph one complete cycle of the given function.

31. Example 4: Graphing a Function of the Form y = AcosBx (6 of 6)
Determine the amplitude and period of
Then graph the function for
Step 5: Extend the graph to the left or right as desired.

32. The Graph of y = Acos(Bx − C)
is obtained by horizontally shifting the graph of y = A cos Bx so that the starting point of the circle is shifted from x = 0 to If
the shift is to the right. If
the shift is to the left. The number
is called the phase shift.

33. Example 5: Graphing a Function of the Form y = Acos(Bx − C) (1 of 6)
Determine the amplitude, period, and phase shift of
Then graph one period of the function.
Step 1: Identify the amplitude, the period, and the phase shift.

34. Example 5: Graphing a Function of the Form y = Acos(Bx − C) (2 of 6)
Determine the amplitude, period, and phase shift of
Then graph one period of the function.
Step 2: Find the x-values for the five key points.

35. Example 5: Graphing a Function of the Form y = Acos(Bx − C) (3 of 6)
Step 3: Find the values of y for the five key points.

36. Example 5: Graphing a Function of the Form y = Acos(Bx − C) (4 of 6)
Step 3: Find the values of y for the five key points. [continued]

37. Example 5: Graphing a Function of the Form y = Acos(Bx − C) (5 of 6)
Step 3: Find the values of y for the five key points. [continued]

38. Example 5: Graphing a Function of the Form y = Acos(Bx − C) (6 of 6)
Determine the amplitude, period, and phase shift of
Then graph one period of the function.
Step 4: Connect the five key points with a smooth curve and graph one complete cycle of the given function.

39. Vertical Shifts of Sinusoidal Graphs
For sinusoidal graphs of the form
the constant D causes a vertical shift in the graph. These vertical shifts result in sinusoidal graphs oscillating about the horizontal line y = D rather than about the x-axis.
The maximum value of y is
The minimum value of y is

40. Example 6: A Vertical Shift (1 of 5)
Graph one period of the function
Step 1: Identify the amplitude, period, phase shift, and vertical shift.
phase shift: 0
vertical shift: one unit upward

41. Example 6: A Vertical Shift (2 of 5)
Graph one period of the function
Step 2: Find the values of x for the five key points.

42. Example 6: A Vertical Shift (3 of 5)
Step 3: Find the values of y for the five key points.

43. Example 6: A Vertical Shift (4 of 5)
Step 3: Find the values of y for the five key points. [continued]

44. Example 6: A Vertical Shift (5 of 5)
Graph one period of the function
Step 4: Connect the five key points with a smooth curve and graph one complete cycle of the given function.

45. Graphing the Sum of Two Trigonometric Functions
We can graph the sum of two (or more) trigonometric functions using a method that involves adding y-coordinates. For example, to graph y = sin x + cos x, we begin by graphing
in the same rectangular coordinate system. Then we select several values of x at which we add the y-coordinates of sin x and cos x. Plotting the points (x, sin x + cos x) and joining them with a smooth curve gives the graph of the desired function.

46. Example 7: Graphing the Sum of Two Trigonometric Functions (1 of 3)
Use the method of adding y-coordinates to graph

47. Example 7: Graphing the Sum of Two Trigonometric Functions (2 of 3)
Use the method of adding y-coordinates to graph

48. Example 7: Graphing the Sum of Two Trigonometric Functions (3 of 3)
Use the method of adding y-coordinates to graph
Plot the nine points from the table and connect with a smooth curve.

49. Example 8: Modeling Periodic Behavior (1 of 5)
A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form
to model the hours
of daylight.
Because the hours of daylight range from a minimum of 10 to a maximum of 14, the curve oscillates about the middle value, 12 hours. Thus, D = 12.

50. Example 8: Modeling Periodic Behavior (2 of 5)
A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form
to model the hours
of daylight.
The maximum number of hours of daylight is 14, which is 2 hours more than 12 hours. Thus, A, the amplitude, is 2; A = 2.

51. Example 8: Modeling Periodic Behavior (3 of 5)
A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form
to model the hours
of daylight.
One complete cycle occurs over a period of 12 months.

52. Example 8: Modeling Periodic Behavior (4 of 5)
The starting point of the cycle is March, x = 3.
The phase shift is

53. Example 8: Modeling Periodic Behavior (5 of 5)
A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form
to model the hours of daylight.
The equation that models the hours of daylight is